ANCHORAGE, Alaska (CNN) -- The investigation into Alaska Gov. Sarah Palin's dismissal of a state official moved forward on two fronts Wednesday.

Palin's husband, Todd, said in a sworn affidavit released Wednesday night that he never pressured the state's public safety commissioner to fire Palin's former brother-in-law, a state trooper who had divorced the governor's sister.

And Alaska's Supreme Court heard arguments Wednesday on an appeal by Gov. Palin's allies to block the Legislature's probe but did not immediately hand down a decision.

Public Safety Commissioner Walt Monegan was not fired this summer because he wouldn't dismiss the trooper, Todd Palin said in his written response to questions from the state Legislature's investigation into Monegan's firing.

"My concerns ... were not why Monegan was reassigned," Palin said in the 52-page document.

Upon the dismissal, Monegan was offered a position as executive director of the Alcohol Beverage and Control Board, but turned it down.

Palin said his wife was unhappy with Monegan because of "budget issues and failure to fill trooper vacancies."

"Walt Monegan was, to the best of my knowledge, not terminated," Palin said. "He was offered a reassignment, and he resigned rather than accept the reassignment."

Monegan has said he believes he was fired because he resisted pressure to terminate the governor's ex-brother-in-law, State Trooper Mike Wooten.

Palin had been resisting a subpoena by lawmakers looking into Monegan's firing since mid-September. But with the report of Stephen Branchflower, the former Anchorage prosecutor hired to conduct the inquiry, expected Friday, Todd Palin agreed to answer written questions submitted through his lawyer.

Sarah Palin, the GOP vice presidential nominee, has denied any wrongdoing, calling Wooten a "rogue trooper" who had threatened her family during his divorce from her sister.

Todd Palin said he relayed the family's concerns to Monegan, but he never urged that the trooper be fired.

"As Monegan has stated, I never told him to fire Wooten," Palin said in his written response to questions in the state Legislature's investigation into Monegan's dismissal.

"My understanding was that he was in charge of receiving any kind of complaint about a trooper. That was his job. At no time did Monegan tell me he felt 'pressure' to do anything he did not think was right."

Todd Palin said his complaints with Monegan centered more on his wife's inability to sometimes use the Department of Public Safety's King Air Turbo Prop plane to reach far-flung constituents in the large state. He and his wife discussed the problem, he said.

Don't Miss

"It seemed like whenever Sarah needed this plane, it was unavailable," Palin said in the document.

"Our discussion was generally that Monegan didn't seem to understand how difficult it was to reschedule these trips on short notice and his failure to get on top of this issue was preventing Sarah from traveling to out-of-the-way rural bush communities to visit her constituents."

Palin said he and his wife thought the department "was retaliating against Sarah" because she had sold a jet that "Department of Safety officials enjoyed using."

The governor's husband said he also contacted Monegan in the "summer of 2007 when a friend's husband drowned in Lake Louise and she was upset with the way the body recovery was being handled. We called Monegan."

In the Alaska Supreme Court front, one of the judges said after hearing argument by lawyers for and against the appeal: "We'll take the case under advisement and adjourn."

The Republican allies of Palin are seeking an end to the investigation. Five GOP lawmakers sued to block the investigation in August, after Palin became Sen. John McCain's running mate.

A judge in Anchorage dismissed their suit last week, ruling that the Legislature could manage its own investigation and upholding subpoenas for members of Palin's administration.

In their appeal, Palin's allies argued the investigation is a Democratic-led witch hunt.

"We are not taking the position that this court should rule that the Legislature cannot, under any circumstances, conduct this investigation," Kevin Clarkson, a lawyer for the GOP lawmakers who filed suit, told the court Wednesday.

"Our point is that this investigation, as it has been conducted to date, has been conducted unconstitutionally and that there is no harm that will tip the scale in their direction if this investigation that is unconstitutional is stopped so that it can be done in a constitutionally compliant manner."

But a lawyer arguing for the probe rejected any suggestion that it be delayed or killed.

"Never in the darkest days of the McCarthy hearings did a court ever say this investigation has to stop, you have to halt what you're doing right now," he said.

"What they're asking for here is not just stopping an investigation, but suppressing a report -- suppressing a report because it might be critical of the governor."

He added, "I think some people would say the more information the better, not just in non-election years, but in election years as well."

Though the probe has had the support of key Republicans in the Legislature, the GOP members who brought the case accuse two Democrats -- state Sens. Kim Elton, the chairman of the Legislative Council, and Hollis French, the Judiciary Committee chairman overseeing the probe -- of trying to damage the Republican presidential ticket.

French in particular has been criticized for a September 2 interview with ABC News in which he warned that the probe could result in an "October Surprise" for the McCain-Palin campaign.

Palin aides have said the governor will cooperate with a separate investigation by the state Personnel Board. Since she became McCain's running mate, her advisers have argued that that agency is the proper venue for the inquiry.